1. **Problem statement:** Find the first three derivatives of the function $$f(x) = \frac{1}{5} (e^x - e^{-x})$$. 2. **Recall the derivative rules:** - The derivative of $$e^x$$ is $$e^x$$. - The derivative of $$e^{-x}$$ is $$-e^{-x}$$ (chain rule). - Constants multiply through the derivative. 3. **First derivative $$f'(x)$$:** $$f'(x) = \frac{1}{5} \left( \frac{d}{dx} e^x - \frac{d}{dx} e^{-x} \right) = \frac{1}{5} (e^x - (-e^{-x})) = \frac{1}{5} (e^x + e^{-x})$$ 4. **Second derivative $$f''(x)$$:** $$f''(x) = \frac{1}{5} \left( \frac{d}{dx} e^x + \frac{d}{dx} e^{-x} \right) = \frac{1}{5} (e^x - e^{-x})$$ 5. **Third derivative $$f'''(x)$$:** $$f'''(x) = \frac{1}{5} \left( \frac{d}{dx} e^x - \frac{d}{dx} e^{-x} \right) = \frac{1}{5} (e^x + e^{-x})$$ **Summary:** - $$f'(x) = \frac{1}{5} (e^x + e^{-x})$$ - $$f''(x) = \frac{1}{5} (e^x - e^{-x})$$ - $$f'''(x) = \frac{1}{5} (e^x + e^{-x})$$ Notice the pattern: the derivatives alternate between $$\frac{1}{5}(e^x - e^{-x})$$ and $$\frac{1}{5}(e^x + e^{-x})$$.